Why wouldn't they hold the supplemental draft immediately after the regular draft? Before UDFAs are allowed to sign...wouldn't that make it much more valuable to those teams who qualified? Or am I missing something or, in a roger clemens-esque term, "mis- thinking" the purpose of the sup draft?

Usually the supplemental is in July, once the players not allowed in the draft (for various reasons) have filed for admission into the league. I don't think eligibility has been determined yet so no clue who or when it is yet